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16 Sep 2014, 00:56
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If points \(A\), \(B\), and \(C\) form a triangle, is angle \(ABC \gt 90\) degrees?
(1) \(AC = AB + BC - 0.001\)
(2) \(AC = AB\)
(1) \(AC = AB + BC - 0.001\)
(2) \(AC = AB\)
16 Sep 2014, 00:56
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Official Solution:
Points A, B and C form a triangle. Is ABC > 90 degrees?
To help determine whether the given triangle has an angle greater than 90 degrees, we begin with some basic triangle properties. Let \(a\), \(b\), and \(c\) be the lengths of the sides of the triangle, where \(c\) is the length of the longest side.
For a right triangle, the Pythagorean theorem states that \(a^2 + b^2 = c^2\).
For an acute triangle (i.e., one with all angles less than 90 degrees), \(a^2 + b^2 > c^2\).
For an obtuse triangle (i.e., one with an angle greater than 90 degrees), \(a^2 + b^2 < c^2\).
(1) AC = AB + BC - 0.001.
If AC = 0.001, AB = 0.001 and BC = 0.001, then the triangle is equilateral, and each angle is 60 degrees.
If AC = 10, AB = 5 and BC = 5.001, then \(AC^2 > AB^2 + BC^2\), which implies that angle ABC (the angle opposite the longest side AC) is greater than 90 degrees..
Not sufficient.
(2) AC = AB:
This statement implies that ABC is an isosceles triangle and angles B and C are equal. If angle B were greater than 90 degrees, then the sum of angles B and C would be greater than 180 degrees, which is not possible. Therefore, angle B cannot be greater than 90 degrees. Sufficient.
Answer: B
Points A, B and C form a triangle. Is ABC > 90 degrees?
To help determine whether the given triangle has an angle greater than 90 degrees, we begin with some basic triangle properties. Let \(a\), \(b\), and \(c\) be the lengths of the sides of the triangle, where \(c\) is the length of the longest side.
For a right triangle, the Pythagorean theorem states that \(a^2 + b^2 = c^2\).
For an acute triangle (i.e., one with all angles less than 90 degrees), \(a^2 + b^2 > c^2\).
For an obtuse triangle (i.e., one with an angle greater than 90 degrees), \(a^2 + b^2 < c^2\).
(1) AC = AB + BC - 0.001.
If AC = 0.001, AB = 0.001 and BC = 0.001, then the triangle is equilateral, and each angle is 60 degrees.
If AC = 10, AB = 5 and BC = 5.001, then \(AC^2 > AB^2 + BC^2\), which implies that angle ABC (the angle opposite the longest side AC) is greater than 90 degrees..
Not sufficient.
(2) AC = AB:
This statement implies that ABC is an isosceles triangle and angles B and C are equal. If angle B were greater than 90 degrees, then the sum of angles B and C would be greater than 180 degrees, which is not possible. Therefore, angle B cannot be greater than 90 degrees. Sufficient.
Answer: B
07 Oct 2015, 10:26
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Statement 2 :- If triangle ABC is isosceles right angle triangle at B ...then statement B is inavalid...for rest other triangle it is valid
some time yes and sometime no
So answer will be E here i guess
some time yes and sometime no
So answer will be E here i guess
....i was missing the key point...here 
